GMAT- Problem Solving 250 Questions and their solutions

Examine how an isosceles right triangle with hypotenuse eight yields equal legs of eight over root two, then apply the area formula to compute the area.

Solve a GMAT geometry problem by comparing shaded regions in four squares with areas 50, 32, 18, and 12, using differences to obtain a 1:3 ratio.

Solve a geometry ratio problem by comparing triangle area to rectangle area using base times height over two, yielding a 1/2 ratio.

solve a geometry problem from gmat problem solving by setting up linear equations from angle relationships, including supplementary angles, to find x=25 and y=33.

On the coordinate plane, line g has a positive slope and never enters the first quadrant, so statements 1 and 3 could be true.

Explore using the point-slope form to derive a line's equation and test multiple points, identifying the point that cannot lie on the line.

Use the difference of squares to equate shaded area with rectangle area and determine width as Y minus X, memorizing Y^2 minus X^2 equals (Y minus X)(Y plus X).

Locate A and B on the x-axis and C above it, determine base AB and C's height (ordinate), then apply triangle area equals base times height divided by two.

Calculate the triangle area from the given coordinates using base times height. Base equals six units; height equals ten units; area equals 30.

Solve a GMAT geometry problem by analyzing line k bounded by the x and y axes with x-intercept 5 and y-intercept 12, using area 30 to determine the slope.

Determine the equation of a line from a system of points, using slope-intercept form and parallel line reasoning, arriving at y = (2/5)x + 2.

Apply the slope formula to points (-1,0) and (2,9) on line K, then find x3 for (x3,21) so the slope remains 3, yielding x3 = 6.

Solve a geometry problem about a line through the origin, determine k from distance ratios and gradient using cross multiplication to obtain -24.5.

Find the x-intercept of the line 3y=2x+6 by setting y=0; solve to get x=-3, so point a on the x-axis is (-3, 0).

Solve geometry problem by analyzing two distinct lines with positive slope and positive y-intercept, identifying that their intersection can lie in quadrants I, II, or III.

Explore geometry problem solving by applying Pythagoras and similar triangles to relate x and y, and derive the shaded area as sqrt(3)/2 x^2 minus y^2.

Apply Pythagoras to a right triangle with x terms, set up a quadratic, solve for x, and recognize 3-4-5 and 5-12-13 triples to confirm x = 3.

From x-intercept −2 and y-intercept 3, use intercept form x/−2 + y/3 = 1 to derive the line and convert to −3x + 2y = 6.

Determine that if two cubes have volumes in a 1:8 ratio, their side lengths are in a 1:2 ratio, so their surface areas are in a 1:4 ratio.

Guide to solving a GMAT geometry problem by relating circle and square through radius and side length, applying Pythagoras and 45‑45‑90 logic, then subtracting circle area to obtain shaded area.

Identify an equilateral triangle with side length six, apply 30-60-90 triangle properties to confirm angles and side ratios, then compute the area as 9 sqrt(3).

Apply 30-60-90 triangle ratios to relate sides to a circle, deducing diameter 8, radius 4, and area 16π using area = πr².

Compute coordinates in a square using 30–60–90 triangles, applying side ratios 1:√3:2 to derive x and y values and locate the point.

learn how to solve a regular hexagon problem by using the exterior angle sum of 360 degrees, interior angles of 120 degrees, and isosceles triangle properties to find key angles.

Apply the Pythagorean theorem to a right triangle to solve for x, then compare x squared to 49 and 64 to conclude 7 < x < 8.

Use symmetry and a 30-60-90 triangle to deduce each side is 12, then sum to get the perimeter of ABCDE as 60.

Solve a coordinate geometry problem by analyzing x- and y-intercepts to determine which equation fits the given constraints, testing options step by step.

Solve for x in a geometry figure with parallel lines. Apply the triangle interior angles sum to 180 degrees and angle correspondences to determine x equals 75 degrees.

solve a geometry problem on squares and similar shapes using area ratios, where the larger square has nine times the smaller's area, to find the smaller side given area 25.

Examine a geometry problem on a large square subdivided into smaller squares and triangles to compute the total shaded area, arriving at 24 cm².

Solve a geometry angle problem using supplementary angle relationships to determine x when angle cft is 85 degrees, yielding x = 130 degrees.

Rank the points by their distance from the origin using the distance formula sqrt(x^2 + y^2). Compare x^2 + y^2 values to get the order P, R, Q.

A regular pentagon yields exterior angles summing to 360, giving interior angles of 108 degrees. The solution finds X as 36° and A as 54°.

Apply the circle circumference formula using a right triangle; derive diameter as sqrt(5) x and compute circumference as pi sqrt(5) x, noting the x must lie outside the radical.

Given a circle with area 9π, determine the area of the equilateral triangle using 30-60-90 triangle ratios and the area formula s^2 sqrt(3) / 4.

Tackle a GMAT geometry problem by applying a 45-45-90 triangle, circle diameter, and side relations to prove the inner square area equals two x squared.

Express w in terms of x and y by applying the triangle angle sum to the triangles, showing how the angles yield a relation among w, x, and y.

Given a rectangle with perimeter 20 and diagonal 9, solve for area using (a+b)^2 = a^2+b^2+2ab and Pythagoras, yielding area = 19/2.

Solve a geometry problem on a circle with diameter 10, where a point inside forms an angle greater than 90, using Pythagoras with AC = 6 to bound BC.

Solve a geometry problem using parallel lines, isosceles properties, and angle relationships to determine x in a quadrilateral, where 70-degree angles lead to x = 80 degrees.

Solve GMAT geometry by comparing areas: identify the blue region where y > x inside a 4 by 3 rectangle, compute triangle area 9/2, and form probability 3/8.

This GMAT problem solving lecture demonstrates using classic right triangles 3-4-5, 5-12-13, and 7-24-25 to compute areas of triangles and a rectangle, yielding a total area of 160.

This lecture analyzes a sphere contained by a cylinder, identifying the complete contact set: two touch points at the top and bottom and a central circle.

The lecture shows the area of an equilateral triangle is (√3/4) a^2, so a 50% side increase yields a 125% area increase.

Learn to solve a geometry angle problem through angle chasing, using equal angles and a straight-line sum to find x, which equals 45.

Compute the area of a right triangle ABC using coordinates on the xy-plane, deducing side lengths from equal y-values and applying base times height over two.

Count all triangles formed by a grid of equally spaced points, considering any orientation; use row-based counting and arithmetic series to arrive at 120.

Solve geometry problems by applying isosceles triangle properties and the angle sum of a triangle to derive expressions for x and y in a GMAT problem.

In a cylinder where diameter equals height, reducing every linear dimension by 60% lowers the volume by 93.6% (approximately 94%).

Explore counting right triangles on an 11 by 11 dot grid, derive 720 triangles by analyzing rectangles and orientations, and apply combinatorial reasoning for GMAT problem solving.

Determine the fraction of the circle’s area that lies outside the equilateral triangle by subtracting the combined areas of two equal equilateral-triangle regions and a 60-degree circular segment from the circle. Use pi r^2 for the circle and sqrt(3)/4 for the equilateral-triangle area to derive the shaded region.

In this GMAT geometry lecture, compute the ratio of the shaded lens region to the triangle using a semicircle, radii, and pi, deriving areas from pi r^2 and triangle area.

Compute the probability that a random chord AB on a circle of radius 2 has length greater than 2.

Three equal circles are tangent to each other and to the sides of an equilateral triangle of side two. A 30-60-90 relation gives r = (√3 − 1)/2.

Count all triangles from 15 evenly spaced circle points by choosing three, totaling 455. Subtract the five equilateral cases to obtain 450 non-equilateral triangles.

analyze a geometry problem with radius six and a 60-degree sector to determine the shaded area using circle-segment and equilateral-triangle calculations.

Use the 1/2 ab sin theta area formula with sides 6 and 8; area equals 24 sin theta, ranging from 0 to 24, with 90 degrees giving 24.

Explore a GMAT geometry problem by determining the shaded region inside a square and rhombus, computing rectangle and triangle areas via 3-4-5 and other Pythagorean triples.

Analyze a circle with center (-2,1) and radius 6, using (x+2)^2+(y-1)^2=36 to count interior and boundary lattice points, yielding 109 points.

Determine the coordinates of points with unit spacing, establishing the point at (-3, 4), and conclude the answer is option B: minus three and four.

Apply geometry in a rectangular coordinate system; a circle tangent to the axes with radius two, use Pythagoras to find ALBE, concluding the answer is B.

Using a 30-60-90 triangle and isosceles properties, this lecture derives side lengths such as 6, 8, and 16 to compute the perimeter of quadrilateral ABCDE.

Identify a right triangle with legs 5 and 12 and hypotenuse 13. Use radius 5 to compute area pi r^2 = 25, concluding the answer is C.

The lecture explains how to compute the coordinates of the midpoint using the formula ((x1+x2)/2, (y1+y2)/2) with x1, y1, x2, y2.

Compute the shaded region between the inscribed circle and the origin in a square, using radius x1, with the shaded area equal to x1^2 minus (pi/4) x1^2.

Determine the ratio of circle area centered at point B to triangle ABC for angle ABC not a multiple of 30, using pi r^2 and 1/2 AB sin angle ABC.

Determine the range of the distance between a point on circle A (radius 12) and a point on circle B (radius 13), showing minimum zero and maximum 50.

Compute circle areas by applying pi r^2. Observe that doubling the radius from 10 to 20 feet increases the area from 100 pi to 400 pi.

Determine radii from equal areas and a given circumference: with circle x's circumference of 16 pi, deduce Rx = 8 and, since areas are equal, Ry = 8.

Use right angles and parallel lines to show similar triangles, then apply area ratios to find AE and the area of triangle AC. The area is 4.5.

This word problem examines a retailer's 80% markup on wholesale price and shows that a 30% drop in selling price corresponds to a 26% increase in wholesale price.

Determine the number of New York Mets fans by modeling three-team fan ratios (Yankees to Mets 3:2, Mets to Red Sox 4:5) and using a total of 300 fans.

Analyze a word problem distributing 50 high schools by type (public, parochial, private independent) across districts A, B, C, with district C equalizing types to solve for private counts.

Convert 90 kilometers per hour to 25 meters per second, then use the distance equals velocity times time formula to compute the 600-meter travel time.

Use the 90,000 overall average for 32 employees, subtract the 560,000 and 880,000 from executives and engineers, then divide by four to get 26,000 for the vice presidents.

Solve a two-step word problem on discounts: 25 percent Monday and 50 percent Tuesday, yielding a regular price of $160 from a final price of $60.

Solve a GMAT problem about micro current in a delicate circuit by dividing 3.6×10^-8 A by 1000 to obtain 3.6×10^-11 A.

Compute the minimum wins needed to reach 60 percent for the year-end tournament, given 14 wins in 18 and 30 games remaining; five more wins are required.

Compute the overall share of employees without an MBA by applying gender-specific MBA rates (30% men, 50% women) and a 40% female workforce, yielding 62% without an MBA.

Solve a system of linear equations to determine the chocolate bar price from gumball and lollipop costs, yielding c = 0.71 dollars.

Explore a GMAT price problem: tariffs kept a console price stable, then a 50% increase; use ratios and cross-multiplication to find the original price before the hike.

Solve a GMAT word problem about three friends funding a gift, using algebra to set up equations with one fourth and one third of the gift price.

Distribute 100 candies among 10 children with at least one each and all distinct; assign 1–9 to nine children to maximize the last child’s share at 55.

Combine the pumping rates of two pumps, A and B, from times 3 and 2 hours to empty the pool, yielding 5/6 pool per hour and 75 minutes.

Solve a word problem by setting six widgets equal to eight widgets at a $1.25 reduction, solve for price, and find Nina’s money as $30.

Solve a four-person age problem by linking Ben's threefold age to Ron and applying the seven-year gaps to Jack and Ken, then use the total 161 to find Ron's age.

Solve a linear word problem: first kilometer costs 85 dollars, each additional costs 5 dollars, total 365 dollars, yielding 57 kilometers.

Combine machine a at 350 widgets per hour and machine b at 250 widgets per hour to determine the time to produce 1000 widgets, which is 1 hour 40 minutes.

Solve a word problem by setting up 3x + 400 = 3(x - 900) to model population changes, finding x = 1550 and inferring Bolivia's current equation.

Solve a word problem with five kids and one dog where the dog's weight is three times the kids' average, yielding the dog's share of the total weight as 3/8.

Use a proportional relationship between marble volume and water rise to solve the word problem: 12 marbles raise 1.5 inches, find how many marbles raise 2.75 inches, yielding 22 marbles.

Explore a GMAT word problem on mixing black and white paint to achieve gray shade using a 2.16:1 ratio and cross-multiplication to estimate white paint for 350 liters of black.

Apply a van diagram to a fifty-student problem with 31 French, 17 Spanish, and 10 neither, and determine that eight study both languages.

Calculate the weekly allowance by following fractions spent: three-fifths at the arcade, one-third of the remaining at the toy store, and zero point eighty dollars at the candy store.

Use weighted averages to relate total weight and number of students, solving 50(B+G)=60B+48G to find B:G=1:5.

Solve a GMAT problem about a barrel's capacity, where the barrel is initially one-fifth full, and the added liquid K yields V in terms of K as V = 15K/7.

Calculate compound interest with semiannual compounding by approximating a 3.96% annual rate to 4% and evaluating $10,000 over two years, yielding about $10,816.

Solve a gmat word problem about population ratios by setting the rest as 4x and the total as 5x, showing Cape Town is 1/5 of the total, or 20%.

Solve a GMAT word problem where apples are four times as many as oranges, with 15% of apples and 10% of oranges growing into trees, yielding x = 600.

Compute the television advertising spend by allocating 60 percent of total revenue (100x) to ads, then taking five sixth of that budget to equal 50x.

Solve a two-part distance problem: a 120-mile trip in six hours with speeds of 25 mph and 50 mph. Ellen travels 75 miles at 25 mph.

Compute the total salaries for 3 vice presidents, 15 executives at 80,000, and 82 data-entry employees at 25,000, using the average of 44,500 across 100 to find the VP average.

This lecture analyzes a GMAT problem on five million dollars at seven percent over two years, showing daily compounding yields the largest total, followed by monthly, quarterly, and annual.

Apply the distance equals velocity times time principle to a GMAT problem: convert eight kilometers per hour to meters per minute, then multiply by six minutes to get 800 meters.

Analyze how a price per transaction and the number of transactions increase by given percent to compute the percent change in last year’s revenue.

Solve a two-segment GMAT problem: a plane travels 120 km/h for 5 hours and 180 km/h thereafter, with a 180 km/h overall average, to find total trip duration.

Explore calculating the probability that the first heart appears on the third draw or later when drawing with replacement from a 52-card deck, using independence and complement.

This GMAT problem solving lecture analyzes a word problem: travel from A to B at 5 km/h, return to achieve a 6 km/h average, using distance equals velocity times time.

Calculate the average velocity of a 100-mile trip where P percent travels at 30 mph and the rest at 50 mph, using total distance over total time.

Start with four quarts of alcohol and four quarts of water, then apply cross-multiplication to obtain a three-to-five alcohol-to-water ratio. Add 8/3 quarts of water to reach the target mixture.

Solve a GMAT problem on distance, speed, and time: determine the usual speed X for a 280 km bus, late by 30 minutes, boosted to 7/6 of normal.

Calculate the percentage of all students who walked by computing 40% of 35 and 80% of 45, then divide the total by 80 to obtain 62.5%.

Solve a two-leg trip with speeds 115 km/h and 135 km/h, total time 5 hours, giving 2.7 hours or 162 minutes.

Analyze how wholesale price relates to retail price in gmat problem solving and show that a 30% markdown yields a final price of 126% of wholesale, i.e., a 26% increase.

Calculate Sara's total: $47 shoes, $16 earrings, and a $25 dance ticket, with 8% tax on the shoes and earrings, giving a final amount of $93.04.

Compare Peter's 100,000 at 12% simple interest with Martha's 100,000 at 12% monthly compounding, showing end-year balances and that Martha earns about 682.50 more.

Set the pigs to cows to chickens ratio as 7:8:10 with a total of 300, and conclude there are 120 chickens.

Determine the price per collection p needed to net profit z, given w total wholesale cost for j crates each with q gift boxes.

Determine how many positive integers under 100 leave remainder 2 when divided by 30 by incrementing by 30 from a base, identify 93 as the largest, and choose option C.

Apply percent growth to a mutual fund problem by using a 25% increase from 2004 to 2005, yielding 5000, then another 25% rise in 2006 gives 6250.

Compute the wholesale price per cup from a bulk order of 80 cups at $692, compare it to a single $12.50 cup, and determine the price difference per cup.

Solve a GMAT word problem linking orchestra and balcony revenues with prices 50 and 30, set up two equations, and derive B in terms of R, yielding B = 500/(300+2R).

Solve a GMAT ratio problem by comparing eight minutes for 30 potatoes to one hour, using cross multiplication and simplification, yielding 225 potatoes; memorize square numbers to speed calculations.

Giving a fraction F of her salary to her husband, a woman invests it at rate R to fund two years, yielding F = 2/(R+3).

Solve a GMAT problem by calculating a 20% price increase from $80 to $96 and applying a 10% employee discount to reach $86.40.

Learn to solve a two-pump pool problem by accounting for a one-minute head start, combining rates 1/a and 1/b, and deriving total time t = (ab+1)/(a+b).

Solve a GMAT word problem on monthly compounding: invest 30,700 at 6.2% annual interest, compounded monthly for three years, and approximate the last month's interest.

Explain a GMAT word problem on distributing money among mom, Lynn, Bob, and Chloe, where the first share is four dollars plus half of what remains, yielding Bob's twenty dollars.

Analyze a GMAT word problem about professor demographics, using percentages for women, men, tenured and non-tenured to determine what percent of men are tenured.

Analyze two cars traveling from town X to town Y at different speeds using passing events and equal distances. Use a GMAT walkthrough applying distance-time equations to solve for speeds.

Guide through a GMAT problem solving word problem about distributing questions across three hours, using fractions to determine the total number of questions and conclude with option b.

Solve a GMAT word problem on half-life. Start with 624 grams; after two 17-month periods, 312 and then 156 grams remain, so 156 grams decayed.

This problem compares trains: A at 60 mph leaves at 3:30 pm; B at 75 mph starts 40 minutes later and overtakes A at 6:50 pm using relative speed.

Solve a word problem about filling a rectangular tank by calculating volume and time. Volume equals 12×10×5 cubic feet; time equals volume divided by 8, giving 75 minutes.

Compute the volume as length times width times depth, then divide by the fill rate eight cubic feet per minute to obtain 75 minutes.

Test divisors of 300 to form equal-sized groups, confirming 100, 75, 60, and 50 work, while 45 cannot divide evenly.

Convert three hours to minutes and multiply the rate of 24 gallons per minute to find the pool capacity of 4,320 gallons.

Master GMAT problem solving by computing monthly revenue from truffles, multiplying price per box by boxes sold, then comparing January and February revenues using a simple multiplication method.

Compute the time by using a constant rate: 600,000 gallons per 4 hours equals 150,000 gallons per hour, so 1,500,000 gallons take 10 hours.

Analyze a GMAT word problem about shipping costs, identifying a fixed first-ounce charge plus a per-additional-ounce rate. Determine the total price as r plus a times (p minus 1).

Compute the immigrant share in a class of 70: 12 immigrant freshmen (40% of 30) and 2 immigrant sophomores (5% of 40) total 14, which is 20% of the class.

Calculate the overall defect rate for a phone factory with 30% model A and 70% model B, where 20% of A and 25% of B are defective, yielding 23.5% defective.

Use the distance equals speed times time approach on a round trip where return speed is half the outbound speed, yielding a total travel time of 2 hours.

A GMAT problem uses a ratio of 1 in 75; with 48 people heard of Brand A, cross multiplication gives a total survey size of 3,600.

In 2001, 300 deer were tagged and released. In 2002, 500 deer were caught and 20 were from the previous year, so 300/P = 20/500, giving P = 7500.

GMAT problem solving: model the number of books read per week using averages, apply total equals average times weeks, and determine a total of 12 books.

Solve a word problem with a Venn diagram to find the percent who took both finance and marketing classes, given 25% finance, 50% marketing, and 40% neither, equal to 15%.

Solve a GMAT word problem: the sum from -35 to an end term equals 150, using the arithmetic series formula and the first term, last term, number of terms.

Determine the overlap of blue flakes and Redflex given 60% blue and 55% Redflex. The portion with both colors is 15%.

Solve a GMAT word problem where price drop by five dollars increases unit sales by ten, keeping revenue at $100; yields 40 units originally and 50 units after the change.

Analyze a GMAT problem involving desktops and laptops, RAM distribution, and conditional percentages to conclude that about 75% of desktops have more than one gigabyte of RAM.